4 HAAR MEASURE ON A QUOTIENT GROUP 269
s e G there exists a complex number x(s) ¥> 0 such that, for every function h e
we have
-> • z) dp,(z) = X(S) jh(z
The measure p, is said to be invariant under G if x(s) = 1 for all 5 e G. In this case the
linear form v :/V-M f^(z) d^(z) is a relatively left-invariant measure on G (Problem 1).
Deduce that x is a continuous homomorphism of G into C* and that for each £ e H
we have x(£) = Ao(£)/AH(£)- (For £ e H, consider the function A (*)=/(*£) on G,
and calculate v(/i) in two ways.) Conversely, if there exists a continuous homomor-
phism x : G -*C* which extends the homomorphism f h-> AG(£)/AH(f) of H into C*,
then there exists a measure ^0 on G/H which is relatively invariant under G. (Prove
thatif /e Jf (G) is such that/b = 0, then we must have fx(x- *)/(*) <*a(*) = 0; to this
end, observe that for each function g e JT(G) we have
and use (a) above and tho theorem of Lebesgue-Fubini.)
(c) In particular, if H is unimodular, there exist nonzero measures on G/H which are
relatively invariant under G. If there exists such a measure ju. on G/H which is positive
and bounded, show that p is G-invariant and that G is unimodular. (If G' is the kernel
of AG , show that H <= G' and that the canonical image of JLL on G/G' is a Haar measure
on this group, and is bounded; deduce that G/G' is compact, and finally that G' = G.)
3. Consider the general linear group GL(2, R), which may be identified with the set of all
matrices
with determinant 7^0. The topology induced from that of R4 is compatible with the
group structure (12.8.1) and makes GL(2, R) a locally compact group. Show that the
measure p. induced on GL(2, R) by Lebesgue measure on R4" is relatively invariant
(Problem 1), and deduce that |det(^T)|"2-^ is a left and right Haar measure on
GL(2, R).
Let T(2, R)* be the subgroup of GL(2, R) consisting of all matrices of the form
Show that a left Haar measure on this group is given by the formula
/W/{//(*>y, t)\x~2t-*\ dx dy dt
and that the modulus function on this group is the function Z\-^\x'"1t\. Deduce that
there exists no relatively GL(2, R)-invariant measure on the homogeneous space
GL(2, R)/T(2, R)* (which can be identified with the projective line PA(R)). (Use the
fact that the commutator subgroup of GL(2, R) is SL(2, R).)
4. Let G be a locally compact group and H a closed subgroup of G such that there exists
a positive bounded G-invariant measure p, ^ 0 on G/H. If U is any relatively compact
open neighborhood of e in G, show that there exists an integer m 2> 1 such that, for. We have An(p5(A)) £ Art(A) (Section 14.1, Problem